Typically, information to be transmitted in a communication system is arranged in relatively large blocks of data before being encoded and transmitted. A data block could comprise e.g. some hundreds or thousands of bits. Such a block is typically encoded in a transmitting node, by use of a code, which is known to a receiving node. The encoded block is then decoded in a receiving node using the known code.
In some applications, it is sometimes desired to convey information consisting of e.g. only one bit, not belonging to a regular block of e.g. payload data. Such an occasional additional lonely bit is then multiplexed with, e.g. inserted between, coded blocks of data. The long blocks of data are typically protected by a powerful error-correcting code, while an additional lonely bit is transmitted separately using a repetition code. For example, in LTE, HARQ ACKs, which may consist of one bit, are multiplexed with blocks of payload data.
An additional lonely bit can be jointly encoded with a data block, and thus also be encoded using e.g. an error correcting code. However, there are at least two reasons for that an additional lonely bit, such as the HARQ ACK bit in LTE, is not jointly encoded with blocks of e.g. payload data. Firstly, it may be desired to detect the lonely bit without decoding the whole data block, which is computationally costly. Secondly, it may be desired to protect the additional lonely bit against error more heavily than the block of payload data, so that the lonely bit can be detected with a smaller error probability than the BER (Block Error Rate) of the payload data.
The problem of multiplexing an additional lonely bit with blocks of payload data is that it is costly to protect the lonely bit against channel noise. This can be understood and quantified as follows. On a (real-valued) AWGN channel with noise variance No/2 and transmit power P (per dimension), bits can be conveyed at a spectral efficiency of:
                    β        =                              1            2                    ⁢                                    log              2                        ⁡                          (                              1                +                                  P                                                            N                      0                                        /                    2                                                              )                                ⁢                                          ⁢          bits          ⁢                      /                    ⁢          channel          ⁢                                          ⁢          use                                    (        1        )            
Beta represents the number of information bits that may be conveyed per symbol or channel use. If K out of N channel uses are used for transmitting an additional lonely bit by use of repetition coding, where the additional lonely bit is repeated K times, the spectral efficiency of the payload transmission would be reduced to:
                              β          ′                =                                            1              2                        ⁢                          (                              1                -                                  K                  N                                            )                        ⁢                                          log                2                            ⁡                              (                                  1                  +                                      P                                                                  N                        0                                            /                      2                                                                      )                                              <          β                                    (        2        )            
In order to restore spectral efficiency from β′ to β, the transmit power P would need to be increased, to P′, where P′ satisfies:
                    β        =                              1            2                    ⁢                      (                          1              -                              K                N                                      )                    ⁢                                    log              2                        (                          1              +                                                P                  ′                                                                      N                    0                                    /                  2                                                      )                                              (        3        )            
Solving (1) and (3) for P and P′, respectively, as functions of β, and computing the ratio P′/P yields:
                              Δ          P                ⁢                  =          Δ                ⁢                                            P              ′                        P                    =                                                    2                                                      2                    ⁢                    β                                                        1                    -                                          K                      /                      N                                                                                  -              1                                                      2                                  2                  ⁢                  β                                            -              1                                                          (        4        )            
Delta P, ΔP, represents the increase in transmit power which is needed to maintain the same spectral efficiency as in an original system, when introducing an additional lonely bit, encoded with a repetition code, which consumes payload resources. Equivalently, in decibel:
                              10          ⁢                                          ⁢                                    log              10                        ⁡                          (                              Δ                P                            )                                      =                                            10              ⁢                                                          ⁢                                                log                  10                                (                                                      2                                                                  2                        ⁢                        β                                                                    1                        -                                                  K                          /                          N                                                                                                      -                  1                                )                                      -                          10              ⁢                                                          ⁢                                                log                  10                                ⁡                                  (                                                            2                                              2                        ⁢                        β                                                              -                    1                                    )                                                              ≈                                    6.0              ·                              1                                  1                  -                                      2                                                                  -                        2                                            ⁢                      β                                                                                  ·                              K                N                                      ⁢                                                  ⁢            dB                                              (        5        )            
Thus, for example, for N=200, K=10, and a target spectral efficiency of β=0.5 bpcu, the power increase, ΔP, needed in order to maintain the same spectral efficiency (as before introduction of an additional bit) is 0.3 dB. It should be noted that these formulas are only asymptotically valid (N→∞ and K<<N) and represent approximations for finite N.
A question of interest is whether there is any way of conveying the additional lonely bit without incurring the power cost of ΔP.